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In some cases, it may be impossible to find an exact function for an antiderivative of a function.
Instead, it might be useful to use approximation methods instead to approximate definite integrals.
In this note, we will revisit the concept of a Riemann sum and how we can use it approximate integrals, then we will soon expand this idea into the concept of a midpoint and trapezoidal approximation. We will also cover the Simpson’s rule for approximating integrals. Finally, we will also explain what error bounds are and how they affect our approximations.
What is a Riemann Sum?
Suppose that we have a function
Now that we know tools for integrating functions, we can simply use the Fundamental Theorem of Calculus to compute the integral. By finding an antiderivative of
However, what if we do not have an exact equation for
{photo of f(x) area under curve [a, b]}
To compute this area, we can divide the interval into equal sections.
Then construct rectangles based on the length of each section and the height of the function at its boundary.
Notice that depending on the function, we overestimate or underestimate some area.
However, by using smaller sections, and thus using more rectangles, we end up approximating the true area of the